The very numerous examples of spiral conformation which we meet with in our studies of organic form are peculiarly adapted to math­e­mat­i­cal methods of in­ves­ti­ga­tion. But ere we begin to study them, we must take care to define our terms, and we had better also attempt some rough preliminary clas­si­fi­ca­tion of the objects with which we shall have to deal.

In general terms, a Spiral Curve is a line which, starting from a point of origin, continually diminishes in curvature as it recedes from that point; or, in other words, whose radius of curvature continually increases. This definition is wide enough to include a number of different curves, but on the other hand it excludes at least one which in popular speech we are apt to confuse with a true spiral. This latter curve is the simple Screw, or cylindrical Helix, which curve, as is very evident, neither starts from a definite origin, nor varies in its curvature as it proceeds. The “spiral” thickening of a woody plant-cell, the “spiral” thread within an insect’s tracheal tube, or the “spiral” twist and twine of a climbing stem are not, math­e­mat­i­cally speaking, spirals at all, but screws or helices. They belong to a distinct, though by no means very remote, family of curves. Some of these helical forms we have just now treated of, briefly and parenthetically, under the subject of Geodetics.

Of true organic spirals we have no lack?495. We think at once of the beautiful spiral curves of the horns of ruminants, and of the still more varied, if not more beautiful, spirals of molluscan shells. Closely related spirals may be traced in the arrangement {494} of the florets in the sunflower; a true spiral, though not, by the way, so easy of in­ves­ti­ga­tion, is presented to us by the outline of a cordate leaf; and yet again, we can recognise typical though transitory spirals in the coil of an elephant’s trunk, in the “circling {495} spires” of a snake, in the coils of a cuttle-fish’s arm, or of a monkey’s or a chameleon’s tail.

Among such forms as these, and the many others which we might easily add to them, it is obvious that we have to do with things which, though math­e­mat­i­cally similar, are biologically speaking fundamentally different. And not only are they biologically remote, but they are also physically different, in regard to the nature of the forces to which they are severally due. For in the first place, the spiral coil of the elephant’s trunk or of the chameleon’s tail is, as we have said, but a transitory configuration, and is plainly the result of certain muscular forces acting upon a structure of a definite, and normally an essentially different, form. It is rather a position, or an attitude, than a form, in the sense in which we have been using this latter term; and, unlike most of the forms which we have been studying, it has little or no direct relation to the phenomenon of Growth.

Again, there is a manifest and not unimportant difference between such a spiral conformation as is built up by the separate and successive florets in the sunflower, and that which, in the snail or Nautilus shell, is apparently a single and indivisible unit. And a similar, if not identical difference is apparent between the Nautilus shell and the minute shells of the Foraminifera, which so closely simulate it; inasmuch as the spiral shells of these latter are essentially composite structures, combined out of successive and separate chambers, while the molluscan shell, though it may (as in Nautilus) become secondarily subdivided, has grown as one continuous tube. It follows from all this that there cannot {496} possibly be a physical or dynamical, though there may well be a math­e­mat­i­cal Law of Growth, which is common to, and which defines, the spiral form in the Nautilus, in the Globigerina, in the ram’s horn, and in the disc of the sunflower.

Of the spiral forms which we have now mentioned, every one (with the single exception of the outline of the cordate leaf) is an example of the remarkable curve known as the Logarithmic Spiral. But before we enter upon the mathematics of the logarithmic spiral, let us carefully observe that the whole of the organic forms in which it is clearly and permanently exhibited, however different they may be from one another in outward appearance, in nature and in origin, nevertheless all belong, in a certain sense, to one particular class of conformations. In the great majority of cases, when we consider an organism in part or whole, when we look (for instance) at our own hand or foot, or contemplate an insect or a worm, we have no reason (or very little) to consider one part of the existing structure as older than another; through and through, the newer particles have been merged and commingled, by intussusception, among the old; the whole outline, such as it is, is due to forces which for the most part are still at work to shape it, and which in shaping it have shaped it as a whole. But the horn, or the snail-shell, is curiously different; for in each of these, the presently existing structure is, so to speak, partly old and partly new; it has been conformed by successive and continuous increments; and each successive stage of growth, starting from the origin, remains as an integral and unchanging portion of the still growing structure, and so continues to represent what at some earlier epoch constituted for the time being the structure in its entirety.

In a slightly different, but closely cognate way, the same is true of the spirally arranged florets of the sunflower. For here again we are regarding serially arranged portions of a composite structure, which portions, similar to one another in form, differ in age; and they differ also in magnitude in a strict ratio according to their age. Somehow or other, in the logarithmic spiral the time-element always enters in; and to this important fact, full of curious biological as well as math­e­mat­i­cal significance, we shall afterwards return. {497}

It is, as we have so often seen, an essential part of our whole problem, to try to understand what distribution of forces is capable of producing this or that organic form,—to give, in short, a dynamical expression to our descriptive morphology. Now the general distribution of forces which lead to the formation of a spiral (whether logarithmic or other) is very easily understood; and need not carry us beyond the use of very elementary mathematics.

If we imagine growth to act in a perpendicular direction, as for example the upward force of growth in a growing stem (OA), then, in the absence of other forces, elongation will as a matter of course proceed in an unchanging direction, that is to say the stem will grow straight upwards. Suppose now that there be some constant external force, such as the wind, impinging on the growing stem; and suppose (for simplicity’s sake) that this external force be in a constant direction (AB) perpendicular to the intrinsic force of growth. The direction of actual growth will be in the line of the resultant of the two forces: and, since the external force is (by hypothesis) constant in direction, while the internal force tends always to act in the line of actual growth, it is obvious that our growing organism will tend to be bent into a curve, to which, for the time being, {498} the actual force of growth will be acting at a tangent. So long as the two forces continue to act, the curve will approach, but will never attain, the direction of AB, perpendicular to the original direction OA. If the external force be constant in amount the curve will ap­prox­i­mate to the form of a hyperbola; and, at any rate, it is obvious that it will never tend to assume a spiral form.

In like manner, if we consider a horizontal beam, fixed at one end, the imposition of a weight at the other will bend the beam into a curve, which, as the beam elongates or the weight increases, will bring the weighted end nearer and nearer to the vertical. But such a force, constant in direction, will obviously never curve the beam into a spiral,—a fact so patent and obvious that it would be superfluous to state it, were it not that some naturalists have been in the habit of invoking gravity as the force to which may be attributed the spiral flexure of the shell.

But if, on the other hand, the deflecting force be inherent in the growing body, or so connected with it in a system that its direction (instead of being constant, as in the former case) changes with the direction of growth, and is perpendicular (or inclined at some constant angle) to this changing direction of the growing force, then it is plain that there is no such limit to the deflection from the normal, but the growing curve will tend to wind round and round its point of origin. In the typical case of the snail-shell, such an intrinsic force is manifestly present in the action of the columellar muscle.

Many other simple illustrations can be given of a spiral course being impressed upon what is primarily rectilinear motion, by any steady deflecting force which the moving body carries, so to speak, along with it, and which continually gives a lop-sided tendency to its forward movement. For instance, we have been told that a man or a horse, travelling over a great prairie, is very apt to find himself, after a long day’s journey, back again near to his starting point. Here some small and imperceptible bias, such as might for instance be caused by one leg being in a minute degree longer or stronger than the other, has steadily deflected the forward movement to one side; and has gradually brought the traveller back, perhaps in a circle to the very point from which he set out, {499} or else by a spiral curve, somewhere within reach and recognition of it.

We come to a similar result when we consider, for instance, a cylindrical body in which forces of growth are at work tending to its elongation, but these forces are unsymmetrically distributed. Let the tendency to elongation along AB be of a magnitude proportional to BB?', and that along CD be of a magnitude proportional to DD?'; and in each element parallel to AB and CD, let a parallel force of growth, proportionately intermediate in magnitude, be at work: and let EFF?' be the middle line. Then at any cross-section BFD, if we deduct the mean force FF?', we have a certain positive force at B, equal to Bb, and an equal and opposite force at D, equal to Dd. But AB and CD are not separate

structures, but are connected together, either by a solid core, or by the walls of a tubular shell; and the forces which tend to separate B and D are opposed, accordingly, by a tension in BD. It follows therefore, that there will be a resultant force BG, acting in a direction intermediate between Bb and BD, and also a resultant, DH, acting at D in an opposite direction; and accordingly, after a small increment of growth, the growing end of the cylinder will come to lie, not in the direction BD, but in the direction GH. The problem is therefore analogous to that of a beam to which we apply a bending moment; and it is plain that the unequal force of growth is equivalent to a “couple” which will impart to our structure a curved form. For, if we regard the part ABDC as practically rigid, and the part BB?'D?'D as pliable, this couple {500} will tend to turn strips such as B?'D?' about an axis perpendicular to the plane of the diagram, and passing through an intermediate point F?'. It is plain, also, since all the forces under consideration are intrinsic to the system, that this tendency will be continuous, and that as growth proceeds the curving body will assume either a circular or a spiral form. But the tension which we have here assumed to exist in the direction BD will obviously disappear if we suppose a sufficiently rapid rate of growth in that direction. For if we may regard the mouth of our tubular shell as perfectly extensible in its own plane, so that it exerts no traction whatsoever on the sides, then it will be drawn out into more and more elongated ellipses, forming the more and more oblique orifices of a straight tube. In other words, in such a structure as we have presupposed, the existence or

maintenance of a constant ratio between the rates of extension or growth in the vertical and transverse directions will lead, in general, to the development of a logarithmic spiral; the magnitude of that ratio will determine the character (that is to say, the constant angle) of the spiral; and the spirals so produced will include, as special or limiting cases, the circle and the straight line.

We may dispense with the hypothesis of bending moments, if we simply presuppose that the increments of growth take place at a constant angle to the growing surface (as AB), but more rapidly at A (which we shall call the “outer edge”) than at B, and that this difference of velocity maintains a constant ratio. Let us also assume that the whole structure is rigid, the new accretions solidifying as soon as they are laid on. For example, {501} let Fig. 242 represent in section the early growth of a Nautilus-shell, and let the part ARB represent the earliest stage of all, which in Nautilus is nearly semicircular. We have to find a law governing the growth of the shell, such that each edge shall develop into an equiangular spiral; and this law, accordingly, must be the same for each edge, namely that at each instant the direction of growth makes a constant angle with a line drawn from a fixed point (called the pole of the spiral) to the point at which growth is taking place. This growth, we now find, may be considered as effected by the continuous addition of similar quadrilaterals. Thus, in Fig. 241, AEDB is a quadrilateral with AE, DB parallel, and with the angle EAB of a certain definite

magnitude, =?. Let AB and ED meet, when produced, in C; and call the angle ACE (or xCy) =. Make the angle yCz =angle xCy, =. Draw EG, so that the angle yEG =?, meeting Cz in G; and draw DF parallel to EG. It is then easy to show that AEDB and EGFD are similar quadrilaterals. And, when we consider the quadrilateral AEDB as having infinitesimal sides, AE and BD, the angle ? tends to a, the constant angle of an equiangular spiral which passes through the points AEG, and of a similar spiral which passes through the points BDF; and the point C is the pole of both of these spirals. In a particular limiting case, when our quadrilaterals are all equal as well as similar,—which will be the case when the angle ? (or the angles EAC, etc.) is a {502} right angle,—the “spiral” curve will be a circular arc, C being the centre of the circle.

In the study of such curves as these, then, we speak of the point of origin as the pole (O); a straight line having its extremity in the pole and revolving about it, is called the radius vector; {503} and a point (P) which is conceived as travelling along the radius vector under definite conditions of velocity, will then describe our spiral curve.

Of several math­e­mat­i­cal curves whose form and development may be so conceived, the two most important (and the only two with which we need deal), are those which are known as (1) the equable spiral, or spiral of Archimedes, and (2) the logarithmic, or equiangular spiral.

The former may be illustrated by the spiral coil in which a sailor coils a rope upon the deck; as the rope is of uniform thickness, so in the whole spiral coil is each whorl of the same breadth as that which precedes and as that which follows it. Using its ancient definition, we may define it by saying, that “If a straight line revolve uniformly about its extremity, a point which likewise travels uniformly along it will describe the equable spiral497.” Or, putting the same thing into our more modern words, “If, while the radius vector revolve uniformly about the pole, a point (P) travel with uniform velocity along it, the curve described will be that called the equable spiral, or spiral of Archimedes.” {504}

It is plain that the spiral of Archimedes may be compared to a cylinder coiled up. And it is plain also that a radius (r =OP), made up of the successive and equal whorls, will increase in arithmetical progression: and will equal a certain constant quantity (a) multiplied by the whole number of whorls, or (more strictly speaking) multiplied by the whole angle (?) through which it has revolved: so that r =a?.

But, in contrast to this, in the logarithmic spiral of the Nautilus or the snail-shell, the whorls gradually increase in breadth, and do so in a steady and unchanging ratio. Our definition is as follows: “If, instead of travelling with a uniform velocity, our point move along the radius vector with a velocity increasing as its distance from the pole, then the path described is called a logarithmic spiral.” Each whorl which the radius vector intersects will be broader than its predecessor in a definite ratio; the radius vector will increase in length in geometrical progression, as it sweeps through successive equal angles; and the equation to the spiral will be r =a?^?. As the spiral of Archimedes, in our example of the coiled rope, might be looked upon as a coiled cylinder, so may the logarithmic spiral, in the case of the shell, be pictured as a cone coiled upon itself.

Now it is obvious that if the whorls increase very slowly indeed, the logarithmic spiral will come to look like a spiral of Archimedes, with which however it never becomes identical; for it is incorrect to say, as is sometimes done, that the Archimedean spiral is a “limiting case” of the logarithmic spiral. The Nummulite is a case in point. Here we have a large number of whorls, very narrow, very close together, and apparently of equal breadth, which give rise to an appearance similar to that of our coiled rope. And, in a case of this kind, we might actually find that the whorls were of equal breadth, being produced (as is apparently the case in the Nummulite) not by any very slow and gradual growth in thickness of a continuous tube, but by a succession of similar cells or chambers laid on, round and round, determined as to their size by constant surface-tension conditions and therefore of unvarying dimensions. But even in this case we should have no Archimedean spiral, but only a logarithmic spiral in which the constant angle approximated to 90°. {505}

There are many other specific properties of the logarithmic spiral, so interrelated to one another that we may choose pretty well any one of them as the basis of our definition, and deduce the others from it either by analytical methods or by the methods of elementary geometry. For instance, the equation r =a?^? may be written in the form logr =?loga, or ? =(logr)/(loga), or (since a is a constant), ? =klogr. Which is as much as to say that the vector angles about the pole are proportional to the logarithms of the successive radii; from which circumstance the name of the “logarithmic spiral” is derived.

Let us next regard our logarithmic spiral from the dynamical point of view, as when we consider the forces concerned in the growth of a material, concrete spiral. In a growing structure, let the forces of growth exerted at any point P be a force F acting along the line joining P to a pole O and a force T acting in a direction perpendicular to OP; and let the magnitude of these forces be in the same constant ratio at all points. It follows that the resultant of the forces F and T (as PQ) makes a constant angle with the radius vector. But the constancy of the angle between tangent and radius vector at any point is a fundamental property of the logarithmic spiral, and may be shewn to follow from our definition of the curve: it gives to the curve its alternative name of equiangular spiral. Hence in a structure growing under the above conditions the form of the boundary will be a logarithmic spiral. {506}

In the concrete case of the shell, the distribution of forces will be, originally, a little more complicated than this, though by resolving the forces in question, the system may be reduced to this simple form. And furthermore, the actual distribution of forces will not always be identical; for example, there is a distinct difference between the cases (as in the snail) where a columellar muscle exerts a definite traction in the direction of the pole, and those (such as Nautilus) where there is no columellar muscle, and where some other force must be discovered, or postulated, to account for the flexure. In the most frequent case, we have, as in Fig. 247, three forces to deal with, acting at a point, p:L, acting

in the direction of the tangent to the curve, and representing the force of longitudinal growth; T, perpendicular to L, and representing the organism’s tendency to grow in breadth; and P, the traction exercised, in the direction of the pole, by the columellar muscle. Let us resolve L and T into components along P (namely A?', B?'), and perpendicular to P (namely A, B); we have now only two forces to consider, viz. P-A?'-B?', and A-B. And these two latter we can again resolve, if we please, so as to deal only with forces in the direction of P and T. Now, the ratio of these forces remaining constant, the locus of the point p is an equiangular spiral. {507}

Furthermore we see how any slight change in any one of the forces P, T, L will tend to modify the angle a, and produce a slight departure from the absolute regularity of the logarithmic spiral. Such slight departures from the absolute simplicity and uniformity of the theoretic law we shall not be surprised to find, more or less frequently, in Nature, in the complex system of forces presented by the living organism.

In the growth of a shell, we can conceive no simpler law than this, namely, that it shall widen and lengthen in the same unvarying proportions: and this simplest of laws is that which Nature tends to follow. The shell, like the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence, and may be made the basis of a definition, of the logarithmic spiral.

Such a definition, though not commonly used by mathematicians, has been occasionally employed; and it is one from which the other properties of the curve can be deduced with great ease and simplicity. In math­e­mat­i­cal language it would run as follows: “Any [plane] curve proceeding from a fixed point (which is called the pole), and such that the arc intercepted between this point and any other whatsoever on the curve is always similar to itself, is called an equiangular, or logarithmic, spiral498.”

In this definition, we have what is probably the most fundamental and “intrinsic” property of the curve, namely the property of continual similarity: and this is indeed the very property by reason of which it is peculiarly associated with organic growth in such structures as the horn or the shell, or the scorpioid cyme which is described on p. 502. For it is peculiarly char­ac­ter­is­tic of the spiral of a shell, for instance, that (under all normal circumstances) it does not alter its shape as it grows; each increment is geometrically similar to its predecessor, and the whole, at any epoch, is similar to what constituted the whole at another and an earlier epoch. We feel no surprise when the animal which secretes the shell, or any other animal whatsoever, grows by such {508} symmetrical expansion as to preserve its form unchanged; though even there, as we have already seen, the unchanging form denotes a nice balance between the rates of growth in various directions, which is but seldom accurately maintained for long. But the shell retains its unchanging form in spite of its asymmetrical growth; it grows at one end only, and so does the horn. And this remarkable property of increasing by terminal growth, but nevertheless retaining unchanged the form of the entire figure, is char­ac­ter­is­tic of the logarithmic spiral, and of no other math­e­mat­i­cal curve.

We may at once illustrate this curious phenomenon by drawing the outline of a little Nautilus shell within a big one. We know, or we may see at once, that they are of precisely the same shape; so that, if we look at the little shell through a magnifying glass, it becomes identical with the big one. But we know, on the other hand, that the little Nautilus shell grows into the big one, not by uniform growth or magnification in all directions, as is (though only ap­prox­i­mate­ly) the case when the boy grows into the man, but by growing at one end only.

Though of all curves, this property of continued similarity is found only in the logarithmic spiral, there are very many rectilinear figures in which it may be observed. For instance, as we may easily see, it holds good of any right cone; for evidently, in Fig. 248, the little inner cone (represented in its triangular section) may become identical with the larger one either by magnification all round (as in a), or simply by an increment at one end (as in b); indeed, in the case of the cone, we have yet a third possibility, for the same result is attained when it increases all round, save only at the base, that is to say when the triangular section increases {509} on two of its sides, as in c. All this is closely associated with the fact, which we have already noted, that the Nautilus shell is but a cone rolled up; in other words, the cone is but a particular variety, or “limiting case,” of the spiral shell.

This property, which we so easily recognise in the cone, would seem to have engaged the particular attention of the most ancient mathematicians even from the days of Pythagoras, and so, with little doubt, from the more ancient days of that Egyptian school whence he derived the foundations of his learning499; and its bearing on our biological problem of the shell, though apparently indirect, is yet so close that it deserves our further consideration.

If, as in Fig. 249, we add to two sides of a square a symmetrical L-shaped portion, similar in shape to what we call a “carpenter’s square,” the resulting figure is still a square; and the portion which we have added is called, by Aristotle (Phys. III, 4), a “gnomon.” Euclid extends the term to include the case of any parallelogram500, whether rectangular or not (Fig. 250); and Hero of Alexandria specifically defines a “gnomon” (as indeed Aristotle implicitly defines it), as any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original. Included in this important definition is the case of numbers, considered geometrically; that is to say, the e?d?t???? ??????, which can be translated into form, by means of rows of dots or other signs (cf. Arist. Metaph. 1092b12), or in the pattern of a tiled floor: all according to “the mystical way of {510} Pythagoras, and the secret magick of numbers.” Thus for example, the odd numbers are “gnomonic numbers,” because

which relation we may illustrate graphically s??at???afe?? by the successive numbers of dots which keep the annexed figure a perfect square?501: as follows:

As we have seen throughout our preliminary discussion, the two most important constants (or chief “specific characters,” as the naturalist would say) in any given logarithmic spiral, are (1) the magnitude of the angle of the spiral, or “constant angle,” a, and (2) the rate of increase of the radius vector for any given angle of revolution, ?. Of this latter, the simplest case is when ? =2p, or 360°; that is to say when we compare the breadths, along the same radius vector, of two successive whorls. As our two magnitudes, that of the constant angle, and that of the ratio of the radii or breadths of whorl, are related to one another, we may determine either of them by actual measurement and proceed to calculate the other.

Thus, in the diagram, OC/OE, or EF/BD, or DC/EF, being in each case radii, or diameters, at right angles to one another, are all equal to e?^(p/2)cota. While in like manner, EO/OF, EG/FH, or GO/HO, all equal e?^pcota; and BC/BA, or CO/OB =e?^2pcota.

As soon, then, as we have prepared tables for these values, the determination of the constant angle a in a particular shell becomes a very simple matter.

A complete table would be cumbrous, and it will be sufficient to deal with the simple case of the ratio between the breadths of adjacent, or immediately succeeding, whorls.

Here we have r =e?^2pcota, or logr =loge×2p×cota, from which we obtain the following figures513: {534}

We learn several interesting things from this short table. We see, in the first place, that where each whorl is about three times the breadth of its neighbour and predecessor, as is the case in Nautilus,

the constant angle is in the neighbourhood of 80°; and hence also that, in all the ordinary Ammonitoid shells, and in all the typically spiral shells of the Gastropods514, the constant angle is also a large one, being very seldom less than 80°, and usually between 80° and 85°. In the next place, we see that with smaller angles the apparent form of the spiral is greatly altered, and the very fact of its being a spiral soon ceases to be apparent (Figs. 271, 272). Suppose one whorl to be an inch in breadth, then, if the angle of the spiral were 80°, the {535} next whorl would (as we have just seen) be about three inches broad; if it were 70°, the next whorl would be nearly ten inches, and if it were 60°, the next whorl would be nearly four feet broad. If the angle were 28°, the next whorl would be a mile and a half in breadth; and if it were 17°, the next would be some 15,000 miles broad.

In other words, the spiral shells of gentle curvature, or of small constant angle, such as Dentalium or Nodosaria, are true logarithmic spirals, just as are those of Nautilus or Rotalia: from which they differ only in degree, in the magnitude of an angular constant. But this diminished magnitude of the angle causes the spiral to dilate with such immense rapidity that, so to speak, “it never comes round”; and so, in such a shell as Dentalium, we never see but a small portion of the initial whorl.

Let us return to the problem of how to ascertain, by direct measurement, the spiral angle of any particular shell. The method already employed is only applicable to complete spirals, that is to say to those in which the angle of the spiral is large, and furthermore it is inapplicable to portions, or broken fragments, of a shell. In the case of the broken fragment, it is plain that the determination of the angle is not merely of theoretic interest, but may be of great practical use to the conchologist as being the one and only way by which he may restore the outline of the missing portions. We have a considerable choice of methods, which have been summarised by, and are partly due to, a very careful student of the Cephalopoda, the late Rev. J. F. Blake515. {537}

• Fig. 274.

  (1) The following method is useful and easy when we have a portion of a single whorl, such as to shew both its inner and its outer edge. A broken whorl of an Ammonite, a curved shell such as Dentalium, or a horn of similar form to the latter, will fall under this head. We have merely to draw a tangent, GEH, to the outer whorl at any point E; then draw to the inner whorl a tangent parallel to GEH, touching the curve in some point F. The straight line joining the points of contact, EF, must evidently pass through the pole: and, accordingly, the angle GEF is the angle required. In shells which bear longitudinal striae or other ornaments, any pair of these will suffice for our purpose, instead of the actual boundaries of the whorl. But it is obvious that this method will be apt to fail us when the angle a is very small; and when, consequently, the points E and F are very remote.
  

• Fig. 275. An Ammonite, to shew corrugated surface-pattern.

  Fig. 276.

  (2) In shells (or horns) shewing rings, or other transverse ornamentation, we may take it that these ornaments are set at a constant angle to the spire, and therefore to the radii. The angle (?) between two of them, as AC, BD, is therefore equal to the angle ? between the polar radii from A and B, or from C and D; and therefore BD/AC =e?^?cota, which gives us the angle a in terms of known quantities. {538}

• (3) If only the outer edge be available, we have the ordinary geometrical problem,—given an arc of an equiangular spiral, to find its pole and spiral angle. The methods we may employ depend (1) on determining directly the position of the pole, and (2) on determining the radius of curvature.

The first method is theo­ret­i­cal­ly sim­ple, but dif­fi­cult in prac­tice; for it requires great accuracy in de­ter­min­ing the points. Let AD, DB, be two tan­gents drawn to the curve. Then a circle drawn through the points ABD will pass through the pole O; since the angles OAD, OBE (the sup­ple­ment of OBD), are equal. The point O may be de­ter­mined by the in­ter­sec­tion of two such circles; and the angle DBO is then the angle, a, required.

Of all these methods by which the math­e­mat­i­cal constants, or specific characters, of a given spiral shell may be determined, the only one of which much use has been made is that which Moseley first employed, namely, the simple method of determining {539} the relative breadths of the whorl at distances separated by some convenient vectorial angle (such as 90°, 180°, or 360°).

Very elaborate measurements of a number of Ammonites have been made by Naumann517, by Sandberger518, and by Grabau519, among which we may choose a couple of cases for consideration. In the following table I have taken a portion of Grabau’s determinations of the breadth of the whorls in Ammonites (Arcestes)

intuslabiatus; these measurements Grabau gives for every 45° of arc, but I have only set forth one quarter of these measurements, that is to say, the breadths of successive whorls measured along one diameter on both sides of the pole. The ratio between alternate measurements is therefore the same ratio as Moseley adopted, namely the ratio of breadth between contiguous whorls along a radius vector. I have then added to these observed values the cor­re­spon­ding calculated values of the angle a, as obtained from our usual formula.

There is considerable irregularity in the ratios derived from these measurements, but it will be seen that this irregularity only implies a variation of the angle of the spiral between about 85° and 87°; and the values fluctuate pretty regularly about the mean, which is 86°15?'. Considering the difficulty of measuring the whorls, especially towards the centre, and in particular the difficulty of determining with precise accuracy the position of the pole, it is clear that in such a case as this we are scarcely justified in asserting that the law of the logarithmic spiral is departed from.

In some cases, however, it is undoubtedly departed from. Here for instance is another table from Grabau, shewing the cor­re­spon­ding ratios in an Ammonite of the group of Arcestes tornatus. In this case we see a distinct tendency of the ratios to

increase as we pass from the centre of the coil outwards, and consequently for the values of the angle a to diminish. The case is precisely comparable to that of a cone with slightly curving sides: in which, that is to say, there is a slight acceleration of growth in a transverse as compared with the longitudinal direction.

In a tubular spiral, whether plane or helicoid, the consecutive whorls may either be (1) isolated and remote from one another; or (2) they may precisely meet, so that the outer border of one and the inner border of the next just coincide; or (3) they may overlap, the vector plane of each outer whorl cutting that of its immediate predecessor or predecessors.

Looking, as we have done, upon the spiral shell as being essentially a cone rolled up, it is plain that, for a given spiral angle, intersection or non-intersection of the successive whorls will depend upon the apical angle of the original cone. For the wider the cone, the more rapidly will its inner border tend to encroach on the outer border of the preceding whorl.

But it is also plain that the greater be the apical angle of the cone, and the broader, consequently, the cone itself be, the greater difference will there be between the total lengths of its inner and outer border, under given conditions of flexure. And, since the inner and outer borders are describing precisely the same spiral about the pole, it is plain that we may consider the inner border as being retarded in growth as compared with the outer, and as being always identical with a smaller and earlier part of the latter.

Dispensing with math­e­mat­i­cal formulae, the several conditions may be illustrated as follows:

In the diagrams (Fig. 278), O P?₁ P?₂ P?₃, etc. represents a radius, on which P?₁, P?₂, P?₃, are the points attained by the outer border of the tubular shell after as many entire consecutive revolutions. And P?₁?', P?₂?', P?₃?', are the points similarly intersected by the inner border; OP/OP?' being always =?, which is the ratio of growth, or “cutting-down factor.” Then, obviously, when O P?₁ is less than O P?₂?' the whorls will be separated by an interspace (a); (2) when O P?₁ =O P?₂?' they will be in contact (b), and (3) when O P?₁ is greater than O P?₂?' there will a greater or less extent of overlapping, that is to say of concealment of the surfaces of the earlier by the later whorls (c). And as a further case (4), it is plain that if ? be very large, that is to say if O P?₁ be greater, not only than O P?₂?' but also than O P?₃?', O P?₄?', etc., we shall have complete, or all but complete concealment by the last formed whorl, of the whole of its predecessors. This latter condition is completely attained in Nautilus pompilius, and approached, though not quite attained, in N. umbilicatus; and the difference between these two forms, or “species,” is constituted accordingly by a difference in the value of ?. (5) There is also a final case, not easily distinguishable externally from (4), where P?' lies on {543} the opposite side of the radius vector to P, and is therefore imaginary. This final condition is exhibited in Argonauta.

The limiting values of ? are easily ascertained.

The case in which ?e?^2pcota =1, or -log? =2pcotaloge, is the case represented in Fig. 278, b: that is to say, the particular case, for each value of a, where the consecutive whorls just touch, without interspace or overlap. For such cases, then, we may tabulate the values of ?, as follows:

We see, accordingly, that in plane spirals whose constant angle lies, say, between 65° and 70°, we can only obtain contact between consecutive whorls if the rate of growth of the inner border of the tube be a small fraction,—a tenth or a twentieth—of that of the outer border. In spirals whose constant angle is 80°, contact is attained when the respective rates of growth are, ap­prox­i­mate­ly, as 3 to 1; while in spirals of constant angle from about 85° to 89°, contact is attained when the rates of growth are in the ratio of from about 3/5 to 9/10.

If on the other hand we have, for any given value of a, a value of ? greater or less than the value given in the above table, then we have, respectively, the conditions of separation or of overlap which are exemplified in Fig. 278, a and c. And, just as we have constructed this table of values of ? for the particular case of simple contact between the whorls, so we could construct similar tables for various degrees of separation, or degrees of overlap.

For instance, a case which admits of simple solution is that in which the interspace between the whorls is everywhere a mean proportional between the breadths of the whorls themselves (Fig. 280). {545}

The cor­re­spon­ding values of ? are as follows:

When ? is a very small fraction, that is to say when the rates of growth of the two borders of the tube are very diverse, then will tend towards infinity—tend that is to say towards a condition in which the inner border of the tube never grows at all. This condition is not infrequently approached in nature. The nearly parallel-sided cone of Dentalium, or the widely separated whorls of Lituites, are evidently cases where ? nearly approaches unity in the one case, and is still large in the other, being correspondingly small; while we can easily find cases where is very large, and ? is a small fraction, for instance in Haliotis, or in Gryphaea.

For the purposes of the morphologist, then, the main result of this last general in­ves­ti­ga­tion is to shew that all the various types of “open” and “closed” spirals, all the various degrees of separation or overlap of the successive whorls, are simply the outward expression of a varying ratio in the rate of growth of the outer as compared with the inner border of the tubular shell.

The foregoing problem of contact, or intersection, of the successive whorls, is a very simple one in the case of the discoid shell but a more complex one in the turbinate. For in the discoid shell contact will evidently take place when the retardation of the inner as compared with the outer whorl is just 360°, and the shape of the whorls need not be considered.

As the angle of retardation diminishes from 360°, the whorls will stand further and further apart in an open coil; as it increases beyond 360°, they will more and more overlap; and when the angle of retardation is infinite, that is to say when the true inner edge of the whorl does not grow at all, then the shell is said to be completely involute. Of this latter condition we have a striking example in Argonauta, and one a little more obscure in Nautilus pompilius.

In the turbinate shell, the problem of contact is twofold, for we have to deal with the possibilities of contact on the same side of the axis (which is what we have dealt with in the discoid) and {547} also with the new possibility of contact or intersection on the opposite side; it is this latter case which will determine the presence or absence of an umbilicus, and whether, if present, it will be an open conical space or a twisted cone. It is further obvious that, in the case of the turbinate, the question of contact or no contact will depend on the shape of the generating curve; and if we take the simple case where this generating curve may be considered as an ellipse, then contact will be found to depend on the angle which the major axis of this ellipse makes with the axis of the shell. The question becomes a complicated one, and the student will find it treated in Blake’s paper already referred to.

When one whorl overlaps another, so that the generating curve cuts its predecessor (at a distance of 2p) on the same radius vector, the locus of intersection will follow a spiral line upon the shell, which is called the “suture” by conchologists. It is evidently one of that ensemble of spiral lines in space of which, as we have seen, the whole shell may be conceived to be constituted; and we might call it a “contact-spiral,” or “spiral of intersection.” In discoid shells, such as an Ammonite or a Planorbis, or in Nautilus umbilicatus, there are obviously two such contact-spirals, one on each side of the shell, that is to say one on each side of a plane perpendicular to the axis. In turbinate shells such a condition is also possible, but is somewhat rare. We have it for instance, in Solarium perspectivum, where the one contact-spiral is visible on the exterior of the cone, and the other lies internally, winding round the open cone of the umbilicus521; but this second contact-spiral is usually imaginary, or concealed within the whorls of the turbinated shell. Again, in Haliotis, one of the contact-spirals is non-existent, because of the extreme obliquity of the plane of the generating curve. In Scalaria pretiosa and in Spirula there is no contact-spiral, because the growth of the generating curve has been too slow, in comparison with the vector rotation of its plane. In Argonauta and in Cypraea, there is no contact-spiral, because the growth of the generating curve has been too quick. Nor, of course, is there any contact-spiral in Patella or in Dentalium, because the angle a is too small ever to give us a complete revolution of the spire. {548}

The various forms of straight or spiral shells among the Cephalopods, which we have seen to be capable of complete definition by the help of elementary mathematics, have received a very complicated descriptive nomenclature from the palaeontologists. For instance, the straight cones are spoken of as orthoceracones or bactriticones, the loosely coiled forms as gyroceracones or mimoceracones, the more closely coiled shells, in which one whorl overlaps the other, as nautilicones or ammoniticones, and so forth. In such a succession of forms the biologist sees undoubted and unquestioned evidence of ancestral descent. For instance we read in Zittel’s Palaeontology522: “The bactriticone obviously represents the primitive or primary radical of the Ammonoidea, and the mimoceracone the next or secondary radical of this order”; while precisely the opposite conclusion was drawn by Owen, who supposed that the straight chambered shells of such fossil cephalopods as Orthoceras had been produced by the gradual unwinding of a coiled nautiloid shell523. To such phylogenetic hypotheses the math­e­mat­i­cal or dynamical study of the forms of shells lends no valid support. If we have two shells in which the constant angle of the spire be respectively 80° and 60°, that fact in itself does not at all justify an assertion that the one is more primitive, more ancient, or more “ancestral” than the other. Nor, if we find a third in which the angle happens to be 70°, does that fact entitle us to say that this shell is intermediate between the other two, in time, or in blood relationship, or in any other sense whatsoever save only the strictly formal and math­e­mat­i­cal one. For it is evident that, though these particular arithmetical constants manifest themselves in visible and recognisable differences of form, yet they are not necessarily more deep-seated or significant than are those which manifest themselves only in difference of magnitude; and the student of phylogeny scarcely ventures to draw conclusions as to the relative antiquity of two allied organisms on the ground that one happens to be bigger or less, or longer or shorter, than the other. {549}

At the same time, while it is obviously unsafe to rest conclusions upon such features as these, unless they be strongly supported and corroborated in other ways,—for the simple reason that there is unlimited room for coincidence, or separate and independent attainment of this or that magnitude or numerical ratio,—yet on the other hand it is certain that, in particular cases, the evolution of a race has actually involved gradual increase or decrease in some one or more numerical factors, magnitude itself included,—that is to say increase or decrease in some one or more of the actual and relative velocities of growth. When we do meet with a clear and unmistakable series of such progressive magnitudes or ratios, manifesting themselves in a progressive series of “allied” forms, then we have the phenomenon of “orthogenesis.” For orthogenesis is simply that phenomenon of continuous lines or series of form (and also of functional or physiological capacity), which was the foundation of the Theory of Evolution, alike to Lamarck and to Darwin and Wallace; and which we see to exist whatever be our ideas of the “origin of species,” or of the nature and origin of “functional adaptations.” And to my mind, the math­e­mat­i­cal (as distinguished from the purely physical) study of morphology bids fair to help us to recognise this phenomenon of orthogenesis in many cases where it is not at once patent to the eye; and also, on the other hand, to warn us, in many other cases, that even strong and apparently complex resemblances in form may be capable of arising independently, and may sometimes signify no more than the equally accidental numerical coincidences which are manifested in identity of length or weight, or any other simple magnitudes.

I have already referred to the fact that, while in general a very great and remarkable regularity of form is char­ac­ter­is­tic of the molluscan shell, that complete regularity is apt to be departed from. We have clear cases of such a departure in Pupa, Clausilia, and various Bulimi, where the enveloping cone of the spire is not a right cone but a cone whose sides are curved. It is plain that this condition may arise in two ways: either by a gradual change in the ratio of growth of the whorls, that is to say in the logarithmic spiral itself, or by a change in the velocity of {550} translation along the axis, that is to say in the helicoid which, in all turbinate shells, is superposed upon the spiral. Very careful measurements will be necessary to determine to which of these factors, or in what proportions to each, the phenomenon is due. But in many Ammonitoidea where the helicoid factor does not enter into the case, we have a clear illustration of gradual and marked changes in the spiral angle itself, that is to say of the ratio of growth cor­re­spon­ding to increase of vectorial angle. We have seen from some of Naumann’s and Grabau’s measurements that such a tendency to vary, such an acceleration or retardation, may be detected even in Ammonites which present nothing abnormal to the eye. But let us suppose that the spiral angle increases somewhat rapidly; we shall then get a spiral with gradually narrowing whorls, and this condition is char­ac­ter­is­tic

of Oekotraustes, a subgenus of Ammonites. If on the other hand, the angle a gradually diminishes, and even falls away to zero, we shall have the spiral curve opening out, as it does in Scaphites, Ancyloceras and Lituites, until the spiral coil is replaced by a spiral curve so gentle as to seem all but straight. Lastly, there are a few cases, such as Bellerophon expansus and some Goniatites, where the outer spiral does not perceptibly change, but the whorls become more “embracing” or the whole shell more involute. Here it is the angle of retardation, the ratio of growth between the outer and inner parts of the whorl, which undergoes a gradual change.

In order to understand the relation of a close-coiled shell to one of its straighter congeners, to compare (for example) an {551} Ammonite with an Orthoceras, it is necessary to estimate the length of the right cone which has, so to speak, been coiled up into the spiral shell. Our problem then is, To find the length of a plane logarithmic spiral, in terms of the radius and the constant angle a. In the annexed diagram, if OP be a radius vector, OQ a line of reference perpendicular to OP, and PQ a tangent to the curve, PQ, or seca, is equal in length to the spiral arc OP. And this is practically obvious: for PP?'/PR?' =ds/dr =seca, and therefore seca =s/r, or the ratio of arc to radius vector.

Accordingly, the ratio of l, the total length, to r, the radius vector up to which the total length is to be measured, is expressed by a simple table of secants; as follows:

Putting the same table inversely, so as to shew the total {552} length in whole numbers, in terms of the radius, we have as follows:

Accordingly, we see that (1), when the constant angle of the spiral is small, the spiral itself is scarcely distinguishable from a straight line, and its length is but very little greater than that of its own radius vector. This remains pretty much the case for a considerable increase of angle, say from 0° to 20° or more; (2) for a very considerably greater increase of the constant angle, say to 50° or more, the shell would only have the appearance of a gentle curve; (3) the char­ac­ter­is­tic close coils of the Nautilus or Ammonite would be typically represented only when the constant angle lies within a few degrees on either side of about 80°. The coiled up spiral of a Nautilus, with a constant angle of about 80°, is about six times the length of its radius vector, or rather more than three times its own diameter; while that of an Ammonite, with a constant angle of, say, from 85° to 88°, is from about six to fifteen times as long as its own diameter. And (4) as we approach an angle of 90° (at which point the spiral vanishes in a circle), the length of the coil increases with enormous rapidity. Our spiral would soon assume the appearance of the close coils of a Nummulite, and the successive increments of breadth in the successive whorls would become inappreciable to the eye. The logarithmic spiral of high constant angle would, as we have already seen, tend to become in­dis­tin­guish­able, without the most careful measurement, from an Archimedean spiral. And it is obvious, moreover, that our ordinary methods of {553} determining the constant angle of the spiral would not in these cases be accurate enough to enable us to measure the length of the coil: we should have to devise a new method, based on the measurement of radii or diameters over a large number of whorls.

The geometrical form of the shell involves many other beautiful properties, of great interest to the mathematician, but which it is not possible to reduce to such simple expressions as we have been content to use. For instance, we may obtain an equation which shall express completely the surface of any shell, in terms of polar or of rectangular coordinates (as has been done by Moseley and by Blake), or in Hamiltonian vector notation. It is likewise possible (though of little interest to the naturalist) to determine the area of a conchoidal surface, or the volume of a conchoidal solid, and to find the centre of gravity of either surface or solid524. And Blake has further shewn, with considerable elaboration, how we may deal with the symmetrical distortion, due to pressure, which fossil shells are often found to have undergone, and how we may reconstitute by calculation their original undistorted form,—a problem which, were the available methods only a little easier, would be very helpful to the palaeontologist; for, as Blake himself has shewn, it is easy to mistake a symmetrically distorted specimen of (for instance) an Ammonite, for a new and distinct species of the same genus. But it is evident that to deal fully with the math­e­mat­i­cal problems contained in, or suggested by, the spiral shell, would require a whole treatise, rather than a single chapter of this elementary book. Let us then, leaving mathematics aside, attempt to summarise, and perhaps to extend, what has been said about the general possibilities of form in this class of organisms.

The surface of any shell, whether discoid or turbinate, may be imagined to be generated by the revolution about a fixed axis of a closed curve, which, remaining always geometrically similar to itself, increases continually its dimensions: and, since the rate of growth of the generating curve and its velocity of rotation follow the same law, the curve traced in space by cor­re­spon­ding points {554} in the generating curve is, in all cases, a logarithmic spiral. In discoid shells, the generating figure revolves in a plane perpendicular to the axis, as in Nautilus, the Argonaut and the Ammonite. In turbinate shells, it slides continually along the axis of revolution, and the curve in space generated by any given point partakes, therefore, of the character of a helix, as well as of a logarithmic spiral; it may be strictly entitled a helico-spiral. Such turbinate or helico-spiral shells include the snail, the periwinkle and all the common typical Gastropods.

The generating figure, as represented by the mouth of the shell, is sometimes a plane curve, of simple form; in other and more numerous cases, it becomes more complicated in form and its boundaries do not lie in one plane: but in such cases as these we

may replace it by its “trace,” on a plane at some definite angle to the direction of growth, for instance by its form as it appears in a section through the axis of the heli­coid shell. The gen­er­ating curve is of very various shapes. It is circular in Scalaria or Cyclostoma, and in Spirula; it may be con­si­dered as a seg­ment of a circle in Natica or in Plan­orbis. It is ap­prox­i­mate­ly tri­an­gular in Conus, and rhom­boidal in Solarium or Potam­ides. It is very com­monly more or less elliptical: the long axis of the el­lipse being parallel to the axis of the shell in Oliva and Cypraea; all but per­pen­di­cu­lar to it in many Trochi; and oblique to it in many well-marked cases, such as Sto­ma­tella, La­mel­laria, Si­gar­etus hal­io­toides (Fig. 284) and Haliotis. In Nautilus pom­pi­lius it is ap­prox­i­mate­ly a semi-ellipse, and in N. um­bil­i­catus rather more than a semi-ellipse, the long axis lying in both cases per­pen­di­cu­lar to the axis of the shell525. Its {555} form is seldom open to easy math­e­mat­i­cal ex­pres­sion, save when it is an actual circle or ellipse; but an exception to this rule may be found in certain Am­mo­nites, for­ming the group “Cordati,” where (as Blake points out) the curve is very nearly rep­re­sent­ed by a cardioid, whose equation is r =a(1+cos?).

The generating curve may grow slowly or quickly; its growth-factor is very slow in Dentalium or Turritella, very rapid in Nerita, or Pileopsis, or Haliotis or the Limpet. It may contain the axis in its plane, as in Nautilus; it may be parallel to the axis, as in the majority of Gastropods; or it may be inclined to the axis, as it is in a very marked degree in Haliotis. In fact, in Haliotis the generating curve is so oblique to the axis of the shell that the latter appears to grow by additions to one margin only (cf. Fig. 258), as in the case of the opercula of Turbo and Nerita referred to on p. 522; and this is what Moseley supposed it to do.

The general appearance of the entire shell is determined (apart from the form of its generating curve) by the magnitude of three angles; and these in turn are determined, as has been sufficiently explained, by the ratios of certain velocities of growth. These angles are (1) the constant angle of the logarithmic spiral (a); (2) in turbinate shells, the enveloping angle of the cone, or (taking half that angle) the angle (?) which a tangent to the whorls makes with the axis of the shell; and (3) an angle called the “angle of retardation” (), which expresses the retardation in growth of {556} the inner as compared with the outer part of each whorl, and therefore measures the extent to which one whorl overlaps, or the extent to which it is separated from, another.

The spiral angle (a) is very small in a limpet, where it is usually taken as =0°; but it is evidently of a significant amount, though obscured by the shortness of the tubular shell. In Dentalium it is still small, but sufficient to give the appearance of a regular curve; it amounts here probably to about 30° to 40°. In Haliotis it is from about 70° to 75°; in Nautilus about 80°; and it lies between 80° and 85°, or even more, in the majority of Gastropods.

The case of Fissurella is curious. Here we have, apparently, a conical shell with no trace of spiral curvature, or (in other words) with a spiral angle which approximates to 0°; but in the minute embryonic shell (as in that of the limpet) a spiral convolution is distinctly to be seen. It would seem, then, that what we have to do with here is an unusually large growth-factor in the generating curve, which causes the shell to dilate into a cone of very wide angle, the apical portion of which has become lost or absorbed, and the remaining part of which is too short to show clearly its intrinsic curvature. In the closely allied Emarginula, there is likewise a well-marked spiral in the embryo, which however is still manifested in the curvature of the adult, nearly conical, shell. In both cases we have to do with a very wide-angled cone, and with a high retardation-factor for its inner, or posterior, border. The series is continued, from the apparently simple cone to the complete spiral, through such forms as Calyptraea.

The angle a, as we have seen, is not always, nor rigorously, a constant angle. In some Ammonites it may increase with age, the whorls becoming closer and closer; in others it may decrease rapidly, and even fall to zero, the coiled shell then straightening out, as in Lituites and similar forms. It diminishes somewhat, also, in many Orthocerata, which are slightly curved in youth, but straight in age. It tends to increase notably in some common land-shells, the Pupae and Bulimi; and it decreases in Succinea.

Directly related to the angle a is the ratio which subsists between the breadths of successive whorls. The following table gives a few illustrations of this ratio in particular cases, in addition to those which we have already studied. {557}

For this formula I have worked out the following table. {558}

From this table, by interpolation, we may easily fill in the ap­prox­i­mate values of a, as soon as we have determined the apical angle ? and measured the ratio R; as follows:

We see from this that shells so different in appearance as Cerithium, Solarium, Natica and Planorbis differ very little indeed in the magnitude of the spiral angle a, that is to say in the relative velocities of radial and tangential growth. It is upon the angle ?

that the difference in their form mainly depends: that is to say the amount of longitudinal shearing, or displacement parallel to the axis of the shell.

The enveloping angle, or rather semi-angle (?), of the cone may be taken as 90° in the discoid shells, such as Nautilus and Planorbis. It is still a large angle, of 70° or 75°, in Conus or in Cymba, somewhat less in Cassis, Harpa, Dolium or Natica; it is about 50° to 55° in the various species of Solarium, about 35° in the typical Trochi, such as T. niloticus or T. zizyphinus, and about 25° or 26° in Scalaria pretiosa and Phasianella bulloides; it becomes a very acute angle, of 15°, 10°, or even less, in Eulima, Turritella or Cerithium. The costly Conus gloria-maris, one of the {560} great treasures of the conchologist, differs from its congeners in no important particular save in the somewhat “produced” spire, that is to say in the comparatively low value of the angle ?.

A variation with advancing age of ? is common, but (as Blake points out) it is often not to be distinguished or disentangled from an alteration of a. Whether alone, or combined with a change in a, we find it in all those many Gastropods whose whorls cannot all be touched by the same enveloping cone, and whose spire is accordingly described as concave or convex. The former condition, as we have it in Cerithium, and in the cusp-like spire of Cassis,

Dolium and some Cones, is much the commoner of the two. And such tendency to decrease may lead to ? becoming a negative angle; in which case we have a depressed spire, as in the Cypraeae.

When we find a “reversed shell,” a whelk or a snail for instance whose spire winds to the left instead of to the right, we may describe it math­e­mat­i­cally by the simple statement that the angle ? has changed sign. In the genus Ampullaria, or Apple-snails, inhabiting tropical or sub-tropical rivers, we have a remarkable condition; for in certain “species” the spiral turns to the right, in others to the left, and in others again we have a flattened {561} “discoid” shell; and furthermore we have numerous intermediate stages, on either side, shewing right and left-handed spirals of varying degrees of acuteness527. In this case, the angle ? may be said to vary, within the limits of a genus, from somewhere about 35° to somewhere about 125°.

The angle of retardation () is very small in Dentalium and Patella; it is very large in Haliotis. It becomes infinite in Argonauta and in Cypraea. Connected with the angle of retardation are the various possibilities of contact or separation, in various degrees, between adjacent whorls in the discoid, and between both adjacent and opposite whorls in the turbinated shell. But with these phenomena we have already dealt sufficiently.

Hitherto we have dealt only with univalve shells, and it is in these that all the math­e­mat­i­cal problems connected with the spiral, or helico-spiral, are best illustrated. But the case of the bivalve shell, of Lamellibranchs or of Brachiopods, presents no essential difference, save only that we have here to do with two conjugate spirals, whose two axes have a definite relation to one another, and some freedom of rotatory movement relatively to one another.

The generating curve is particularly well seen in the bivalve, where it simply constitutes what we call “the outline of the shell.” It is for the most part a plane curve, but not always; for there are forms, such as Hippopus, Tridacna and many Cockles, or Rhynchonella and Spirifer among the Brachiopods, in which the edges of the two valves interlock, and others, such as Pholas, Mya, etc., where in part they fail to meet. In such cases as these the generating curves are conjugate, having a similar relation, but of opposite sign, to a median plane of reference. A great variety of form is exhibited by these generating curves among the bivalves. In a good many cases the curve is ap­prox­i­mate­ly circular, as in Anomia, Cyclas, Artemis, Isocardia; it is nearly semi-circular in Argiope. It is ap­prox­i­mate­ly elliptical in Orthis and in Anodon; it may be called semi-elliptical in Spirifer. It is a nearly rectilinear {562} triangle in Lithocardium, and a curvilinear triangle in Mactra. Many apparently diverse but more or less related forms may be shewn to be deformations of a common type, by a simple application of the math­e­mat­i­cal theory of “Transformations,” which we shall have to study in a later chapter. In such a series as is furnished, for instance, by Gervillea, Perna, Avicula, Modiola, Mytilus, etc., a “simple shear” accounts for most, if not all, of the apparent differences.

Upon the surface of the bivalve shell we usually see with great clearness the “lines of growth” which represent the successive margins of the shell, or in other words the successive positions assumed during growth by the growing generating curve; and we have a good illustration, accordingly, of how it is char­ac­ter­is­tic of the generating curve that it should constantly increase, while never altering its geometric similarity.

Underlying these “lines of growth,” which are so char­ac­ter­is­tic of a molluscan shell (and of not a few other organic formations), there is, then, a “law of growth” which we may attempt to enquire into and which may be illustrated in various ways. The simplest cases are those in which we can study the lines of growth on a more or less flattened shell, such as the one valve of an oyster, a Pecten or a Tellina, or some such bivalve mollusc. Here around an origin, the so-called “umbo” of the shell, we have a series of curves, sometimes nearly circular, sometimes elliptical, and often asymmetrical; and such curves are obviously not “concentric,” though we are often apt to call them so, but are always “co-axial.” This manner of arrangement may be illustrated by various analogies. We might for instance compare it to a series of waves, radiating outwards from a point, through a medium which offered a resistance increasing, with the angle of divergence, according to some simple law. We may find another, and perhaps a simpler illustration as follows:

That is to say, all the balls reach the circumference of the circle at the same moment as the ball which drops vertically from A to C.

Where, then, as often happens, the generating curve of the shell is ap­prox­i­mate­ly a circle passing through the point of origin, we may consider the acceleration of growth along various radiants to be governed by a simple math­e­mat­i­cal law, closely akin to that simple law of acceleration which governs the movements of a falling body. And, mutatis mutandis, a similar definite law underlies the cases where the generating curve is continually elliptical, or where it assumes some more complex, but still regular and constant form.

It is easy to extend the proposition to the particular case where the lines of growth may be considered elliptical. In such a case we have x?²/a?²+y?²/b?² =1, where a and b are the major and minor axes of the ellipse.

Obviously, in the case when a =b, this gives us the circular system which we have already considered. For other values, or ratios, of a and b, and for all values of ?, we can easily construct a table, of which the following is a sample:

The coaxial ellipses which we then draw, from the values given in the table, are such as are shewn in Fig. 288 for the ratio a/b =3/1, and in Fig. 289 for the ratio a/b =1/2; these are fair ap­prox­i­ma­tions to the actual out­lines, and to the actual ar­range­ment of the lines of growth, in such forms as Sole­cur­tus or Cul­tel­lus, and in Tel­lina or Psam­mobia. It is not dif­ficult to intro­duce a cons­tant into our equa­tion to meet the case of a shell which is somewhat unsymmetrical on either side of the median axis. It is a somewhat more troublesome matter, however, to bring these con­fi­gur­a­tions into relation with a “law of growth,” as was so easily done in the case of the circular figure: in other words, to {565} formulate a law of acceleration according to which points starting from the origin O, and moving along radial lines, would all lie, at any future epoch, on an ellipse passing through O; and this calculation we need not enter into.

All that we are immediately concerned with is the simple fact that where a velocity, such as our rate of growth, varies with its direction,—varies that is to say as a function of the angular divergence from a certain axis,—then, in a certain simple case, we get lines of growth laid down as a system of coaxial circles, and, when the function is a more complex one, as a system of ellipses or of other more complicated coaxial figures, which figures may or may not be symmetrical on either side of the axis. Among our bivalve mollusca we shall find the lines of growth to be ap­prox­i­mate­ly circular in, for instance, Anomia; in Lima (e.g. L. subauriculata) we have a system of nearly symmetrical ellipses with the vertical axis about twice the transverse; in Solen pellucidus, we have again a system of lines of growth which are not far from being symmetrical ellipses, in which however the transverse is between three and four times as great as the vertical axis. In the great majority of cases, we have a similar phenomenon with the further complication of slight, but occasionally very considerable, lateral asymmetry.

In certain little Crustacea (of the genus Estheria) the carapace takes the form of a bivalve shell, closely simulating that of a {566} lamellibranchiate mollusc, and bearing lines of growth in all respects analogous to or even identical with those of the latter. The explanation is very curious and interesting. In ordinary Crustacea the carapace, like the rest of the chitinised and calcified integument, is shed off in successive moults, and is restored again as a whole. But in Estheria (and one or two other small crustacea) the moult is incomplete: the old carapace is retained, and the new, growing up underneath it, adheres to it like a lining, and projects beyond its edge: so that in course of time the margins of successive old carapaces appear as “lines of growth” upon the surface of the shell. In this mode of formation, then (but not in the usual one), we obtain a structure which “is partly old and partly new,” and whose successive increments are all similar, similarly situated, and enlarged in a continued progression. We have, in short, all the conditions appropriate and necessary for the development of a logarithmic spiral; and this logarithmic spiral (though it is one of small angle) gives its own character to the structure, and causes the little carapace to partake of the char­ac­ter­is­tic conformation of the molluscan shell.

The essential simplicity, as well as the great regularity of the “curves of growth” which result in the familiar con­fi­gur­a­tions of our bivalve shells, sufficiently explain, in a general way, the ease with which they may be imitated, as for instance in the so-called “artificial shells” which Kappers has produced from the conchoidal form and lamination of lumps of melted and quickly cooled paraffin528.

Among the bivalves the spiral angle (a) is very small in the flattened shells, such as Orthis, Lingula or Anomia. It is larger, as a rule, in the Lamellibranchs than in the Brachiopods, but in the latter it is of considerable magnitude among the Pentameri. Among the Lamellibranchs it is largest in such forms as Isocardia and Diceras, and in the very curious genus Caprinella; in all of these last-named genera its magnitude leads to the production of a spiral shell of several whorls, precisely as in the univalves. The angle is usually equal, but of opposite sign, in the two valves of the Lamellibranch, and usually of opposite sign but unequal in the two valves of the Brachiopod. It is very unequal in many Ostreidae, and especially in such forms as Gryphaea, or in Caprinella, which is a kind of exaggerated Gryphaea. Occasionally it is of the same sign in both valves (that is to say, both valves curve the same way) as we see sometimes in Anomia, and much better in Productus or Strophomena.

Owing to the large growth-factor of the generating curve, and the comparatively small angle of the spiral, the whole shell seldom assumes a spiral form so conspicuous as to manifest in a typical way the helical twist or shear which is so conspicuous in the {568} majority of univalves, or to let us measure or estimate the magnitude of the apical angle (?) of the enveloping cone. This however we can do in forms like Isocardia and Diceras; while in Caprinella we see that the whorls lie in a plane perpendicular to the axis, forming a discoidal spire. As in the latter shell, so also universally among the Brachiopods, there is no lateral asymmetry in the plane of the generating curve such as to lead to the development of a helix; but in the majority of the Lamellibranchiata it is obvious, from the obliquity of the lines of growth, that the angle ? is significant in amount.

The so-called “spiral arms” of Spirifer and many other Brachiopods are not difficult to explain. They begin as a single structure, in the form

of a loop of shelly substance, attached to the dorsal valve of the shell, in the neighbourhood of the hinge. This loop has a curvature of its own, similar to but not necessarily identical with that of the valve to which it is attached; and this curvature will tend to be developed, by continuous and symmetrical growth, into a fully formed logarithmic spiral, so far as it is permitted to do so under the constraint of the shell in which it is contained. In various Terebratulae we see the spiral growth of the loop, more or less flattened and distorted by the restraining pressure of the ventral valve. In a number of cases the loop remains small, but gives off two nearly parallel branches or offshoots, which continue to grow. And these, starting with just such a slight curvature as the loop itself possessed, grow on and on till they may form close-wound spirals, always provided that the “spiral angle” of the curve is such that the resulting spire can be freely contained within the cavity of the shell. Owing to the bilateral symmetry of the whole system, the case will be rare, and unlikely to occur, in which each separate arm will coil strictly in a plane, so as to constitute a discoid spiral; for the original {569} direction of each of the two branches, parallel to the valve (or nearly so) and outwards from the middle line, will tend to constitute a curve of double curvature, and so, on further growth, to develop into a helicoid. This is what actually occurs, in the great majority of cases. But the curvature may be such that the helicoid grows outwards from the middle line, or inwards towards the middle line, a very slight difference in the initial curvature being sufficient to direct the spire the one way or the other; the middle course of an undeviating discoid spire will be rare, from the usual lack of any obvious controlling force to prevent its deviation. The cases in which the helicoid spires point towards, or point away from, the middle line are ascribed, in zoological clas­si­fi­ca­tion, to particular “families” of Brachiopods, the former condition defining

(or helping to define) the Atrypidae and the latter the Spiriferidae and Athyridae. It is obvious that the incipient curvature of the arms, and consequently the form and direction of the spirals, will be influenced by the surrounding pressures, and these in turn by the general shape of the shell. We shall expect, accordingly, to find the long outwardly directed spirals associated with shells which are transversely elongated, as Spirifer is; while the more rounded Atrypas will tend to the opposite condition. In a few cases, as in Cyrtina or Reticularia, where the shell is comparatively narrow but long, and where the uncoiled basal support of the arms is long also, the spiral coils into which the latter grow are turned backwards, in the direction where there is room for them. And in the few cases where the shell is very considerably flattened, the spirals (if they find room {570} to grow at all) will be constrained to do so in a discoid or nearly discoid fashion, and this is actually the case in such flattened forms as Koninckina or Thecidium.

While math­e­mat­i­cally speaking we are entitled to look upon the bivalve shell of the Lamellibranch as consisting of two distinct elements, each comparable to the entire shell of the univalve, we have no biological grounds for such a statement; for the shell arises from a single embryonic origin, and afterwards becomes split into portions which constitute the two separate valves. We can perhaps throw some indirect light upon this phenomenon, and upon several other phenomena connected with shell-growth, by a consideration of the simple conical or tubular shells of the Pteropods. The shells of the latter are in few cases suitable for simple math­e­mat­i­cal in­ves­ti­ga­tion, but nevertheless they are of very considerable interest in connection with our general problem.

The morphology of the Pteropods is by no means well understood, and in speaking of them I will assume that there are still grounds for believing (in spite of Boas’ and Pelseneer’s arguments) that they are directly related to, or may at least be directly compared with, the Cephalopoda529.

The simplest shells among the Pteropods have the form of a tube, more or less cylindrical (Cuvierina), more often conical (Creseis, Clio); and this tubular shell (as we have already had occasion to remark, on p. 258), frequently tends, when it is very small and delicate, to assume the character of an unduloid. (In such a case it is more than likely that the tiny shell, or that portion of it which constitutes the unduloid, has not grown by successive {571} increments or “rings of growth,” but has developed as a whole.) A thickened “rib” is often, perhaps generally, present on the dorsal side of the little conical shell. In a few cases (Limacina, Peraclis) the tube becomes spirally coiled, in a normal logarithmic spiral or helico-spiral.

In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is curiously modified. In the first place, its cross-section, originally {572} circular or nearly so, becomes flattened or compressed dorso-ventrally; and the angle, or rather edge, where dorsal and ventral walls meet, becomes more and more drawn out into a ridge or keel. Along the free margin, both of the dorsal and the ventral portion of the shell, growth proceeds with a regularly varying velocity, so that these margins, or lips, of the shell become regularly curved or markedly sinuous. At the same time, growth in a transverse direction proceeds with an acceleration which manifests itself in a curvature of the sides, replacing the straight borders of the original cone. In other words, the cross-section of the cone, or what we have been calling the generating curve, increases its dimensions more rapidly than its distance from the pole.

In the above figures, for instance in that of Cleodora cuspidata, the markings of the shell which represent the successive edges of the lip at former stages of growth, furnish us at once with a “graph” of the varying velocities of growth as measured, radially, from the apex. We can reveal more clearly the nature of these variations in the following way which is simply tantamount to converting our radial into rectangular coordinates. Neglecting curvature (if any) of the sides and treating the shell (for simplicity’s sake) as a right cone, we lay off equal angles from the apex O, along the radii Oa, Ob, etc. If we then plot, as vertical equidistant ordinates, the magnitudes Oa, Ob ... OY, and again on to Oa?', we obtain a diagram such as the following (Fig. 299); by {573} help of which we not only see more clearly the way in which the growth-rate varies from point to point, but we also recognise much better than before, the similar nature of the law which governs this variation in the different species.

Furthermore, the young shell having become differentiated into a dorsal and a ventral part, marked off from one another by a lateral edge or keel, and the inequality of growth being such as to cause each portion

to increase most rapidly in the median line, it follows that the entire shell will appear to have been split into a dorsal and a ventral plate, both connected with, and projecting from, {574} what remains of the original undivided cone. Putting the same thing in other words, we may say that the generating figure, which lay at first in a plane perpendicular to the axis of the cone, has now, by unequal growth, been sharply bent or folded, so as to lie ap­prox­i­mate­ly in two planes, parallel to the anterior and posterior faces of the cone. We have only to imagine the apical connecting portion to be further reduced, and finally to disappear or rupture, and we should have a bivalve shell developed out of the original simple cone.

In its outer and growing portion, the shell of our Pteropod now consists of two parts which, though still connected together at the apex, may be treated as growing practically independently. The shell is no longer a simple tube, or simple cone, in which regular inequalities of growth will lead to the development of a spiral; and this for the simple reason that we have now two opposite maxima of growth, instead of a maximum on the one side and a minimum on the other side of our tubular shell. As a matter of fact, the dorsal and the ventral plate tend to curve in opposite directions, towards the middle line, the dorsal curving ventrally and the ventral curving towards the dorsal side.

In the case of the Lamellibranch or the Brachiopod, it is quite possible for both valves to grow into more or less pronounced spirals, for the simple reason that they are hinged upon one another; and each growing edge, instead of being brought to a standstill by the growth of its opposite neighbour, is free to move out of the way, by the rotation about the hinge of the plane in which it lies.

But where, as in the Pteropod, there is no such hinge, the dorsal and ventral halves of the shell (or dorsal and ventral valves, if we may call them so), if they curved towards one another (as they do in a cockle), would soon interfere with one another’s progress, and the development of a pair of conjugate spirals would become impossible. Nevertheless, there is obviously, in both dorsal and ventral valve, a tendency to the development of a spiral curve, that of the ventral valve being more marked than that of the larger and overlapping dorsal one, exactly as in the two unequal valves of Terebratula. In many cases (e.g. Cleodora cuspidata), the dorsal valve or plate, {575} strengthened and stiffened by its midrib, is nearly straight, while the curvature of the other is well displayed. But the case will be materially altered and simplified if growth be arrested or retarded in either half of the shell. Suppose for instance that the dorsal valve grew so slowly that after a while, in comparison with the other, we might speak of it as being absent altogether: or suppose that it merely became so reduced in relative size as to form no impediment to the continued growth of the ventral one; the latter would continue to grow in the direction of its natural curvature, and would end by forming a complete and coiled logarithmic spiral. It would be precisely analogous to the spiral shell of Nautilus, and, in regard to its

ventral position, concave towards the dorsal side, it would even deserve to be called directly homologous with it. Suppose, on the other hand, that the ventral valve were to be greatly reduced, and even to disappear, the dorsal valve would then pursue its unopposed growth; and, were it to be markedly curved, it would come to form a logarithmic spiral, concave towards the ventral side, as is the case in the shell of Spirula530. Were the dorsal valve to be destitute of any marked curvature (or in other words, to have but a low spiral angle), it would form a simple plate, as in the shells of Sepia or Loligo. Indeed, in the shells of these latter, and especially in that of Sepia, we seem to recognise a manifest resemblance to the dorsal plate of the Pteropod shell, as we have it (e.g.) in Cleodora or Hyalaea; {576} the little “rostrum” of Sepia is but the apex of the primitive cone, and the rounded anterior extremity has grown according to a law precisely such as that which has produced the curved margin of the dorsal valve in the Pteropod. The ventral portion of the original cone is nearly, but not wholly, wanting. It is represented by the so-called posterior wall of the “siphuncular space.” In many decapod cuttle-fishes also (e.g. Todarodes, Illex, etc.) we still see at the posterior end of the “pen,” a vestige of the primitive cone, whose dorsal margin only has continued to grow; and the same phenomenon, on an exaggerated scale, is represented in the Belemnites.

It is not at all impossible that we may explain on the same lines the development of the curious “operculum” of the Ammonites. This consists of a single horny plate (Anaptychus), or of a thicker, more calcified plate divided into two symmetrical halves (Aptychi), often found inside the terminal chamber of the Ammonite, and occasionally to be seen lying in situ, as an operculum which partially closes the mouth of the shell; this structure is known to exist even in connection with the early embryonic shell. In form the Anaptychus, or the pair of conjoined Aptychi, shew an upper and a lower border, the latter strongly convex, the former sometimes slightly concave, sometimes slightly convex, and usually shewing a median projection or slightly developed rostrum. From this “rostral” border the curves of growth start, and course round parallel to, finally constituting, the convex border. It is this convex border which fits into the free margin of the mouth of the Ammonite’s shell, while the other is applied to and overlaps the preceding whorl of the spire. Now this relationship is precisely what we should expect, were we to imagine as our starting-point a shell similar to that of Hyalaea, in which however the dorsal part of the split cone had become separate from the ventral half, had remained flat, and had grown comparatively slowly, while at the same time it kept slipping forward over the growing and coiling spire into which the ventral half of the original shell develops531. In short, I think there is reason to believe, or at least to suspect, that we {577} have in the shell and Aptychus of the Ammonites, two portions of a once united structure; of which other Cephalopods retain not both parts but only one or other, one as the ventrally situated shell of Nautilus, the other as the dorsally placed shell for example of Sepia or of Spirula.

In the case of the bivalve shells of the Lamellibranchs or of the Brachiopods, we have to deal with a phenomenon precisely analogous to the split and flattened cone of our Pteropods, save only that the primitive cone has been split into two portions, not incompletely as in the Pteropod (Hyalaea), but completely, so as to form two separate valves. Though somewhat greater freedom is given to growth now that the two valves are separate and hinged, yet still the two valves oppose and hamper one another, so that in the longitudinal direction each is capable of only a moderate curvature. This curvature, as we have seen, is recognisable as a logarithmic spiral, but only now and then does the growth of the spiral continue so far as to develop successive coils: as it does in a few symmetrical forms such as Isocardia cor; and as it does still more conspicuously in a few others, such as Gryphaea and Caprinella, where one of the two valves is stunted, and the growth of the other is (relatively speaking) unopposed.

Before we leave the subject of the molluscan shell, we have still another problem to deal with, in regard to the form and arrangement of the septa which divide up the tubular shell into chambers, in the Nautilus, the Ammonite and their allies (Fig. 304, etc.).

The existence of septa in a Nautiloid shell may probably be accounted for as follows. We have seen that it is a property of a cone that, while growing by increments at one end only, it conserves its original shape: therefore the animal within, which (though growing by a different law) also conserves its shape, will continue to fill the shell if it actually fills it to begin with: as does a snail or other Gastropod. But suppose that our mollusc fills a part only of a conical shell (as it does in the case of Nautilus); then, unless it alter its shape, it must move upward as it grows in the growing cone, until it come to occupy a space similar in form {578} to that which it occupied before: just, indeed, as a little ball drops far down into the cone, but a big one must stay farther up. Then, when the animal after a period of growth has moved farther up in the shell, the mantle-surface continues its normal secretory activity, and that portion which had been in contact with the former septum secretes a septum anew. In short, at any given epoch, the creature is not secreting a tube and a septum by separate operations, but is secreting a shelly case about its rounded body, of which case one part appears to us as the continuation of the tube, and the other part, merging with it by in­dis­tin­guish­able boundaries, appears to us as the septum532.

The various forms assumed by the septa in spiral shells533 present us with a number of problems of great beauty, simple in their essence, but whose full in­ves­ti­ga­tion would soon lead us into mathematics of a very high order.

We do not know in great detail how these septa are laid down; but the essential facts are clear534. The septum begins as a very thin cuticular membrane (composed apparently of a substance called conchyolin), which is secreted by the skin, or mantle-surface, of the animal; and upon this membrane nacreous matter is gradually laid down on the mantle-side (that is to say between the animal’s body and the cuticular membrane which has been thrown off from it), so that the membrane remains as a thin pellicle over the hinder surface of the septum, and so that, to begin with, the membranous septum is moulded on the flexible and elastic surface of the animal, within which the fluids of the body must exercise a uniform, or nearly uniform pressure.

Moreover, since the cavity below the septum is practically closed, and is filled either with air or with water, P will be constant over the whole area of the septum. And further, we must assume, at least to begin with, that the membrane constituting the incipient septum is homogeneous or isotropic.

Let us take first the case of a straight cone, of circular section, more or less like an Orthoceras; and let us suppose that the septum is attached to the shell in a plane perpendicular to its axis. The septum itself must then obviously be spherical. Moreover the extent of the spherical surface is constant, and easily determined. For obviously, in Fig. 302, the angle LCL?' equals the supplement of the angle (LOL?') of the cone; that is to say, the circle of contact subtends an angle at the centre of the spherical surface, which is constant, and which is equal to p-2?. The case is not excluded where, owing to an asymmetry of tensions, the septum meets the side walls of the cone at other than a right angle,

as in Fig. 303; and here, while the septa still remain portions of spheres, the geometrical construction for the position of their centres is equally easy.

If, on the other hand, the attachment of the septum to the inner walls of the cone be in a plane oblique to the axis, then it is evident that the outline of the septum will be an ellipse, and its surface an {580} ellipsoid. If the attachment of the septum be not in one plane, but form a sinuous line of contact with the cone, then the septum will be a saddle-shaped surface, of great complexity and beauty. In all cases, provided only that the membrane be isotropic, the form assumed will be precisely that of a soap-bubble under similar conditions of attachment: that is to say, it will be (with the usual limitations or conditions) a surface of minimal area.

But while the outline of the septum in median section is simple and easy to determine, the curved surface of the septum in its entirety is a very complicated matter, even in Nautilus which is one of the simplest of actual cases. For, in the first place, since the form of the septum, as seen in median section, is that of a logarithmic spiral, and as therefore its curvature is constantly altering, it follows that, in successive transverse sections, the {581} curvature is also constantly altering. But in the case of Nautilus, there are other aspects of the phenomenon, which we can illustrate, but only in part, in the following simple manner. Let us imagine

a pack of cards, in which we have cut out of each card a similar concave arc of a logarithmic spiral, such as we actually see in the median section of the septum of a Nautilus. Then, while we hold the cards together, foursquare, in the ordinary position of the {582} pack, we have a simple “ruled” surface, which in any longitudinal section has the form of a logarithmic spiral but in any transverse section is a straight horizontal line. If we shear or slide the cards upon one another, thrusting the middle cards of the pack forward in advance of the others, till the one end of the pack is a convex, and the other a concave, ellipse, the cut edges which combine to represent our septum will now form a curved surface

of much greater complexity; and this is part, but not by any means all, of the deformation produced as a direct consequence of the form in Nautilus of the section of the tube within which the septum has to lie. And the complex curvature of the surface will be manifested in a sinuous outline of the edge, or line of attachment of the septum to the tube, and will vary according to the configuration of the latter. In the case of Nautilus, it is easy to shew empirically (though not perhaps easy to demonstrate {583} math­e­mat­i­cally) that the sinuous or saddle-shaped form of the “suture” (or line of attachment of the septum to the tube) is such as can be precisely accounted for in this manner. It is also easy to see that, when the section of the tube (or “generating curve”) is more complicated in form, when it is flattened, grooved, or otherwise ornamented, the curvature of the septum and the outline of its sutural attachment will become very complicated indeed536; but it will be comparatively simple in the case of the first few sutures of the young shell, laid down before any overlapping of whorls has taken place, and this comparative simplicity of the first-formed sutures is a marked feature among Ammonites537.

We have other sources of complication, besides those which are at once introduced by the sectional form of the tube. For instance, the siphuncle, or little inner tube which perforates the septa, exercises a certain amount of tension, sometimes evidently considerable, upon the latter; so that we can no longer consider each septum as an isotropic surface, under uniform pressure; and there may be other structural modifications, or inequalities, in that portion of the animal’s body with which the septum is in contact, and by which it is conformed. It is hardly likely, for all these reasons, that we shall ever attain to a full and particular explanation of the septal surfaces and their sutural outlines throughout the whole range of Cephalopod shells; but in general terms, the problem is probably not beyond the reach of math­e­mat­i­cal analysis. The problem might be approached experimentally, after the manner of Plateau’s experiments, by bending {584} a wire into the complicated form of the suture-line, and studying the form of the liquid film which constitutes the cor­re­spon­ding surface minimae areae.

In certain Ammonites the septal outline is further complicated in another way. Superposed upon the usual sinuous outline, with its “lobes” and “saddles,” we have here a minutely ramified, or arborescent outline,

in which all the branches terminate in wavy, more or less circular arcs,—looking just like the ‘landscape marble’ from the Bristol Rhaetic. We have no difficulty in recognising in this a surface-tension phenomenon. The figures are precisely such as we can imitate (for instance) by pouring a {585} few drops of milk upon a greasy plate, or of oil upon an alkaline solution.

We have very far from exhausted, we have perhaps little more than begun, the study of the logarithmic spiral and the associated curves which find exemplification in the multitudinous diversities of molluscan shells. But, with a closing word or two, we must now bring this chapter to an end.

In the spiral shell we have a problem, or a phenomenon, of growth, immensely simplified by the fact that each successive increment is irrevocably fixed in regard to magnitude and position, instead, of remaining in a state of flux and sharing in the further changes which the organism undergoes. In such a structure, then, we have certain primary phenomena of growth manifested in their original simplicity, undisturbed by secondary and conflicting phenomena. What actually grows is merely the lip of an orifice, where there is produced a ring of solid material, whose form we have treated of under the name of the generating curve; and this generating curve grows in magnitude without alteration of its form. Besides its increase in areal magnitude, the growing curve has certain strictly limited degrees of freedom, which define its motions in space: that is to say, it has a vector motion at right angles to the axis of the shell; and it has a sliding motion along that axis. And, though we may know nothing whatsoever about the actual velocities of any of these motions, we do know that they are so correlated together that their relative velocities remain constant, and accordingly the form and symmetry of the whole system remain in general unchanged.

But there is a vast range of possibilities in regard to every one of these factors: the generating curve may be of various forms, and even when of simple form, such as an ellipse, its axes may be set at various angles to the system; the plane also in which it lies may vary, almost indefinitely, in its angle relatively to that of any plane of reference in the system; and in the several velocities of growth, of rotation and of translation, and therefore in the ratios between all these, we have again a vast range of possibilities. We have then a certain definite type, or group of forms, math­e­mat­i­cally isomorphous, but presenting infinite diversities of outward appearance: which diversities, as Swammerdam {586} said, ex sola nascuntur diversitate gyrationum; and which accordingly are seen to have their origin in differences of rate, or of magnitude, and so to be, essentially, neither more nor less than differences of degree.

In nature, we find these forms presenting themselves with but little relation to the character of the creature by which they are produced. Spiral forms of certain particular kinds are common to Gastropods and to Cephalopods, and to diverse families of each; while outside the class of molluscs altogether, among the Foraminifera and among the worms (as in Spirorbis, Spirographis, and in the Dentalium-like shell of Ditrupa), we again meet with similar and cor­re­spon­ding forms.

Again, we find the same forms, or forms which (save for external ornament) are math­e­mat­i­cally identical, repeating themselves in all periods of the world’s geological history; and, irrespective of climate or local conditions, we see them mixed up, one with another, in the depths and on the shores of every sea. It is hard indeed (to my mind) to see where Natural Selection necessarily enters in, or to admit that it has had any share whatsoever in the production of these varied conformations. Unless indeed we use the term Natural Selection in a sense so wide as to deprive it of any purely biological significance; and so recognise as a sort of natural selection whatsoever nexus of causes suffices to differentiate between the likely and the unlikely, the scarce and the frequent, the easy and the hard: and leads accordingly, under the peculiar conditions, limitations and restraints which we call “ordinary circumstances,” one type of crystal, one form of cloud, one chemical compound, to be of frequent occurrence and another to be rare.